Consider the conjugate Orlicz spaces corresponding to the Young
functions \(\Phi (x)=e^{x}-x-1\)  and \(\Psi (x)=(1+x)\ln (1+x)-x\) ,
their respective norms being denoted by \(||\cdot ||_{\Phi }\) , \(||\cdot ||_{\Psi }\) , see
the Appendix of [1]} for definitions.
Assumption REF  then implies \(\sup _{t\in [0,T]}||S_{t}^{2}||_{\Phi }<\infty \) .
Using Proposition A-2-2 of [1]}, we get that
\(\mathbb {E}[\xi S_{t}^{2}]\le C ||\xi ||_{\Psi }||S_{t}^{2}||_{\Phi } \) 
with some constant \(C\) . The statement follows.
